Self-consistent atmosphere modeling with cloud formation for low-mass stars and exoplanets
AAstronomy & Astrophysics - accepted for publicationAugust 24, 2017
Self-consistent atmosphere modeling with cloud formationfor low-mass stars and exoplanets
Diana Juncher (cid:63) , U ff e G. Jørgensen and Christiane Helling Niels Bohr Institute & Centre for Star and Planet Formation, University of Copenhagen, Øster Voldgade 5, 1350 Copenhagen, DK Centre for Exoplanet Science, SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, FifeKY16 9SS, UK
ABSTRACT
Context.
Low-mass stars and extrasolar planets have ultra-cool atmospheres where a rich chemistry occurs and clouds form. Theincreasing amount of spectroscopic observations for extrasolar planets requires self-consistent model atmosphere simulations to con-sistently include the formation processes that determine cloud formation and their feedback onto the atmosphere.
Aims.
Complement the M arcs model atmosphere suit with simulations applicable to low-mass stars and exoplanets in preparation ofE-ELT, JWST, PLATO and other upcoming facilities.
Methods.
The M arcs code calculates stellar atmosphere models, providing self-consistent solutions of the radiative transfer and the at-mospheric structure and chemistry. We combine M arcs with a kinetic model that describes cloud formation in ultra-cool atmospheres(seed formation, growth / evaporation, gravitational settling, convective mixing, element depletion). Results.
We present a small grid of self-consistently calculated atmosphere models for T e ff = − g ) = .
5. Cloud formation in stellar and sub-stellar atmospheres appears for T e ff < ff ect on the structure and the spectrum of the atmosphere for T e ff < τ =
1) changes with wavelength resulting in a flux variation of ∼ rift -M arcs for an example exoplanet and demonstrate that our simulations reproduce the Spitzer observations for WASP-19brather well for T e ff = g ) = . Key words. astrochemistry – radiative transfer – methods: numerical – stars: atmospheres, low-mass, brown dwarfs
1. Introduction
The atmospheres of late type M-dwarf stars and brown dwarfs- collectively referred to as ultra cool dwarfs - and planets havelow enough temperatures for clouds to form. Cloud formationincreases the total atmospheric opacity but also a ff ects the localgas phase by element depletion. A strong e ff ect on the structureof their atmospheres results, and hence self-consistent inclusionof cloud formation is critical for inferring correctly the physicalstructure and chemical composition of these objects from ob-served spectra. The same physics and considerations apply to theatmospheres of the bulk of known exoplanets, and the present pa-per is therefore our first paper in a series of planned papers to de-scribe self-consistent modeling of exoplanetary atmospheres asa tool for interpreting coming high-quality spectra of exoplanetsthat will become available with the next generation instrumentsduring the coming years.The presence of cloud formation in ultra cool dwarf atmo-spheres was first proposed by Lunine et al. (1986) from the com-parison of temperature-pressure profiles of brown dwarf atmo-sphere models with the condensation curves of refractory mate-rials such as iron, sodium-aluminum silicates, and magnesiumsilicates. A decade later, Tsuji et al. (1996) presented the firstcloud modeling results for brown dwarfs, showing how the highopacity of dust particles can produce a noticeably e ff ect in the (cid:63) e-mail: [email protected] observed spectrum. Tsuji et al. (1996) also suggested that cloudformation should be considered for all objects with T e ff < ff erent coupled processes that depend on a wide range of physi-cal and chemical parameters. Many of the early models of cloudyatmospheres (e.g. Rossow (1978), Lewis (1969), Carlson et al.(1988), Lunine et al. (1986), Burrows et al. (1989), and Tsujiet al. (1996)) were able to reproduce basic features of ultra cooldwarfs by simply turning on or o ff the opacity of dust in theatmosphere at its chemical equilibrium temperature-pressure lo-cation. Over the years the models have grown more detailed andmore realistic, and today several independent groups are workingon complex models that represent clouds in atmosphere modelsusing di ff erent strategies. Some are based on practical consid-erations (Tsuji 2001; Barman et al. 2011; Burrows et al. 2006),while others are inspired by measurements of the atmospheres ofthe planets in our Solar system (Allard et al. 2001; Cooper et al.2003), terrestrial cloud formation (Ackerman & Marley 2001),or kinetic dust-formation modeling in asymptotic giant branchstars (Helling et al. 2001; Woitke & Helling 2003, 2004). A de-tailed comparison between a selection of these can be found inHelling et al. (2008a).In this paper we present an extension of the M arcs code(Gustafsson 1971; Jørgensen et al. 1992; Gustafsson et al. 2008; Article number, page 1 of 17 a r X i v : . [ a s t r o - ph . S R ] A ug & A proofs: manuscript no. aa
Van Eck et al. 2017) that has so far been used extensively formodeling atmospheres of cool stars (Lambert et al. 1986; Plez1992; Aringer et al. 1997), including abundance analysis (e.g.Blackwell et al. (1995); Matrozis et al. (2013); Nissen et al.(2014); Hill et al. (2016); Siqueira-Mello et al. (2016)), H O de-tections (Ryde et al. 2002; Aringer et al. 2002), microdiamondsin carbon stars (Andersen & Jørgensen 1995), and instrumentcalibration (Decin et al. 2003; Decin & Eriksson 2007). M arcs has also been used to study cool, helium-rich white dwarfs (Jør-gensen et al. 2000), R Coronae Borealis stars (Asplund et al.2000), and to determine fundamental properties of GRB progen-itors (Groh et al. 2013). While the radiative-transfer treatment ofM arcs has inspired time-dependent carbon-rich models for dust-forming AGB stars (Höfner et al. 1998), the lower mass counter-part, i.e. late type M-dwarfs and brown dwarfs with clouds, hasnot been addressed by the M arcs community so far. This paperpresents M arcs model atmosphere simulations which include adetailed modeling of cloud formation, by self-consistently solv-ing the radiative transfer and gas-phase chemistry in the schemeof marcs together with the seed formation, growth / evaporationof cloud particles, element conservation and gravitational set-tling in the scheme of drift . In this way the radiative and chem-ical feedback on the atmosphere due to cloud formation is fullytaken into account. Section 2 summarizes our approach, includ-ing tables of input properties. We present our results for a grid ofD rift -M arcs model atmosphere simulations applicable to solar-metallicity M-dwarfs and brown dwarfs ( T e ff = − g ) = .
5; Section 3). These models represent an exten-sion of the M arcs code with respect to the updated gas-phaseopacity data and the modeling of cloud formation. They also of-fer a new alternative to the D rift -P hoenix models. We comparethe synthetic spectra resulting from our atmosphere simulationswith observed spectra of mid- to late-type M-dwarfs, early tomid-type L-dwarfs, and an example giant gas planet WASP-19bin Section 4. Section 5 discusses the e ff ect of porosity in cloudparticles. Appendix B provides additional details about the gasspecies contributing to the synthetic spectra.
2. Approach
Two well-tested codes are combined to enable hands-on atmo-sphere simulations for ultra-cool, cloud-forming objects. D rift ,the cloud formation module, has been applied to investigatecloud structures in brown dwarfs and extrasolar planets fromfirst principles (e.g. Helling et al. (2008b); Street et al. (2015);Helling et al. (2016)). M arcs has been applied to a large numberof atmosphere problems (Section 1). We follow a similar strat-egy as in Witte et al. (2009) in combining the two codes. In thefollowing, we provide a summery of the two codes, the opacitydata used, and the methodology for running the combined codes. M arcs The code:
The M arcs code was introduced in the early 1970sby Gustafsson et al. (1975) and has since then been developedin step with the advancement of computer power and availablephysical data. The most recent general grid of M arcs models waspublished by Gustafsson et al. (2008) and contains about 50,000state-of-the-art stellar atmosphere models extending from late A-type to early M-type stars - from dwarfs to supergiants - for vary-ing metallicities and C / O-ratios. This version of M arcs is verysimilar to our version, and details of the implementation of hy-drostatic equilibrium, radiative transfer, convection and mixing
Table 1.
Sources of data for continuum opacities. "b-f" and "f-f" denotebound-free and free-free processes, respectively. CIA stands for colli-sion induced absorption.
Ion Process ReferenceH − b-f Doughty et al. (1966)H − f-f Doughty & Fraser (1966)H I b-f, f-f Karzas & Latter (1961)H I + H I CIA Doyle (1968)H − f-f Somerville (1964)H + f-f Mihalas (1965)He − f-f Somerville (1965); John (1967)He I f-f Peach (1970)C I , f-f Peach (1970)Mg I f-f Peach (1970)Al I , f-f Peach (1970)Si I f-f Peach (1970)e − scattering Mihalas (1978)H I scattering Dalgarno, quoted by Kurucz (1970)length can be found in Gustafsson et al. (2008). For the equi-librium calculations we use a version of Tsuji’s program (Tsuji1964) implemented by Helling et al. (1996), and updated furtherfor the present work. Input data:
The chemical equilibrium calculations in M arcs are based on 38 atoms and 210 molecules (see Appendix A). Wehave adopted the chemical composition of the Sun as reportedby Grevesse et al. (2007) for all our models. For the atoms andions we use the internal partition function data from Irwin (1981)to calculate the equilibrium constants. For the molecules we usethe Gibbs free energy data from Tsuji (1973); Burrows & Sharp(1999); Burrows et al. (2005) to calculate the equilibrium con-stants.We calculate the continuum absorption for about a dozenions, electron scattering and Rayleigh scattering by H I (Table 1).The line opacities for atoms and ions were updated by Popo-vas (2014) with atomic line data from VALD-3 (Kupka et al.2011). The line opacities for molecules were updated to includethe 24 molecules and molecular pairs listed in Table 2. We sam-pled all line opacities using the Opacity Sampling method witha resolution of R = λ/ ∆ λ = . − µ m.As described in Gustafsson et al. (2008), the convection inM arcs is handled using the mixing length method, where theconvective energy flux can be calculated as a function of the mix-ing length l . The value of l is based on empirical calibrations ofstellar interior models and is thus not theoretically derived. It isoften expressed as a product of the mixing length parameter α and the scale height. For cool stars and brown dwarfs α ≈ D rift The code:
The D rift code models cloud formation by con-sidering each of the involved physical and chemical processesin detail. The formation of seed particles and the subsequentgrowth or evaporation of dust grains are describe by modifiedclassical nucleation theory and the moment method, respectively(Gail & Sedlmayr 1988; Dominik et al. 1993; Lee et al. 2015).The initial model equations where extended to describe the
Article number, page 2 of 17iana Juncher, U ff e G. Jørgensen and Christiane Helling: Atmosphere modeling with cloud formation Table 2.
Molecular line transitions and their sources.Molecule Transitions Reference
Hydrides
LiH vib-rot Coppola et al. (2011)MgH vib-rot Yadin et al. (2012)A-X, B’-X GharibNezhad et al. (2013)SiH A-X Kurucz (2011)CaH vib-rot Yadin et al. (2012)A-X, B-X, C-X, D-X, E-X Weck et al. (2003)TiH A-X, B-X Burrows et al. (2005)CrH A-X Burrows et al. (2002)FeH F-X Wende et al. (2010)CH vib-rot, A-X, B-X, C-X Masseron et al. (2014)NH vib-rot Brooke et al. (2014a)A-X, A-C Kurucz (2011)OH vib-rot, A-X Kurucz (2011)
Oxides
SiO vib-rot Barton et al. (2013)A-X, E-X Kurucz (2011)TiO A-X, B-X, C-X, E-X Schwenke (1998)c-a, b-a, b-d, f-aVO A-X, B-X, C-X Kurucz (2011)ZrO B-A, B-X, C-X, E-A Plez et al. (2003)b-a, d-a, e-a, f-aCO vib-rot, A-X Kurucz (2011)NO vib-rot Rothman et al. (2010)H O vib-rot Jørgensen et al. (2001)
Other H , HD vib-rot, quad, B-X, C-X Kurucz (2011)C A-X, b-a, E-A Kurucz (2011)d-a Brooke et al. (2013)CN vib-rot, A-X, B-X Brooke et al. (2014b)CO vib-rot Rothman et al. (2010)HCN vib-rot Harris et al. (2006)Harris et al. (2008)H -H CIA Borysow et al. (2001)H -He CIA Jørgensen et al. (2000) growth / evaporation of particles of mixed material compositionas required in particular for oxygen-rich atmospheres (Helling& Woitke 2006; Helling et al. 2008b). This is coupled to thee ff ects of gravitational settling, convective mixing and elementdepletion via a system of partial di ff erential equations (Woitke& Helling 2003, 2004; Helling & Woitke 2006).The convection in ultra cool dwarfs allow for the upwardstransport and subsequent di ff usion of the non-depleted gas fromthe interior of the dwarf. This convective mixing can be extendedinto the upper, radiative atmosphere via overshooting, therebyfacilitating a replenishment of the depleted gas above the cloudbase, maintaining the dust cycle. The D rift code models over-shooting by assuming an exponential decrease of the mass ex-change frequency above the radiative zone (Equation 9 in Woitke& Helling (2004), with β = τ minmix = / ( H p v c )).We consider seven growth species (TiO [s], MgSiO [s],SiO [s], Fe[s], Al O [s], MgO[s] and MgSiO [s]) to make thisinitial implementation as simple as possible. We include 32chemical surface reactions which is a subset of reactions ofHelling et al. (2008b) for the respective materials. For TiO weuse the data from Woitke & Helling (2003) to calculate the satu-ration vapor pressure at di ff erent temperatures. For the remainingcondensates we use the data from Sharp & Huebner (1990). Table 3.
References for n and k optical constants of the condensates. Solid species ReferenceTiO [s] Ribarsky in Palik (1985)MgSiO [s] Jäger et al. (2003)SiO [s] Posch et al. (2003)Fe[s] Posch et al. (2003)Al O [s] Zeidler et al. (2013)MgO[s] Roessler & Hu ff man in Palik (1985)MgSiO [s] Dorschner et al. (1995) D ust opacity : D rift calculates the vertical distribution of theclouds as well as the size and composition of their cloud parti-cles, but to assess how the opacity of the clouds a ff ect the struc-ture we also need to calculate the absorption and scattering ofthe dust grains.From the information provided by D rift about a specificcloud particle size and the volume of each of its components, wecan use the Bruggeman Equations (Bruggeman 1935) to calcu-late its e ff ective index of refraction, assuming that the dust grainis compact and its components are randomly mixed. This allowus to treat the dust grain as a homogeneous particle, the proper-ties of its components combining to generate e ff ective propertiesof the whole particle itself.Because the size of the dust grains are typically of the sameorder as the wavelength of the starlight, we cannot use theRayleigh or geometrical approximations to describe how theyinteract with the light. Instead we have to use full Mie Theory(Mie 1908; Bohren & Hu ff man 1983) for a complete descrip-tion of how electromagnetic plane weaves are absorbed and scat-tered by homogeneous spherical particles. This, of course, alsorequires the assumption that the dust grains are spherical. Input Data:
The sources of the optical constants used to calcu-late the e ff ective index of refraction of the mixed dust particlesare given in Table 3. Most of the data covers the wavelengthrange 1 . − µ m, only the data for Al O [s] and MgSiO [s]had to be extrapolated down to the lowest considered wave-length. We did this by freezing the optical constants from thefirst known wavelength points. M arcs with Drift In order to calculate the details of the cloud layers in an at-mosphere, D rift needs information about the ( T g , P g )-structure,chemical composition and convection of the atmosphere. Simi-larly, M arcs needs information about the size and compositionof the cloud particles as well as the depletion of elements to cal-culate the e ff ects of clouds in the atmosphere. We manage thisdata exchange between M arcs and D rift through input and out-put files containing the information listed in Table 4. Changes to the M arcs code: In previous versions of theM arcs code the element abundances have been considered con-stant throughout the atmosphere. Since di ff usion of atoms is avery slow process that only becomes dominant in stars hotterthan T e ff ≈ ,
500 K (Hui-Bon-Hoa et al. 2000), this is usu-ally an excellent approximation and especially so for late typestars, where the deep convective envelopes will keep the gas wellmixed. However, in ultra cool dwarfs the dust formation willcause a depletion of elements in the top layers where the dustgrains form, and a corresponding augmentation of elements in
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Table 4.
The data exchanged between M arcs and D rift . M arcs to D rift D rift to M arcs layer height z a ( z ) average grain sizegas temperature T ( z )gas pressure P ( z ) V i ( z ) average graingas density ρ ( z ) volume fractionsgravitational acceleration g ( z )convection velocity v c ( z ) (cid:15) i ( z ) depleted elementmixing length parameter l abundancesinitial element abundances (cid:15) i the layers where the dust grains evaporates. We have thereforeexpanded the initial one-dimensional array containing elementabundances with an extra dimension to account for their depthdependence.The coolest models of Gustafsson et al. (2008) start at aRosseland optical depth of log( τ ) = −
5, ends at log( τ ) =
2, andhave a resolution of ∆ log( τ ) = .
2. This is appropriate for cloudfree models, and extending the atmospheres or increasing theresolution would have little e ff ect on the computed model. Butsince cloud formation can set in at much lower optical depthsthan log( τ ) = − τ ) = −
10. Furthermore, we also increased the resolutionin the upper layers to ∆ log( τ ) = .
15 to accommodate the po-tentially rapid changes over short distances cloud formation cancause.
Changes to the D rift code: The changes to the D rift codewere minimal, consisting only of the addition of a new routineto handle the communication with M arcs . Running D rift - M arcs : While M arcs needs to know the ele-ment depletion and dust opacity before it can solve the radia-tive transfer equation, D rift needs to know the convection speedwhich is calculated by M arcs as it solves the radiative transferequation. At a first glance it seems like we are in a deadlock,but the solution is actually quite simple. If we start with a dustfree model of T e ff ≈ ff ective temperature, iterating through M arcs and D rift foreach step, the data exchange files will be updated in sync withthe increasing dust formation. For this to work the change inthe e ff ective temperature between each step had to be relativelysmall, about T e ff = −
50 K depending on the impact of thedust formation.The dust free version of M arcs will keep iterating over amodel until the temperature corrections in all layers are belowa given value, usually T ≤ arcs to fulfill this convergence criterion every time we run D rift , wecan easily end up in a endless loop with no convergence in sight.When D rift adds a layer of dust to the atmosphere, M arcs willheat the layers as a reaction to the increased opacity. In response,D rift will then reduce the amount of dust as the higher temper-atures impede the dust formation. M arcs will of course react tothe decreased opacity by cooling the layers again, and we arethus back where we started - or even further away! To avoid this,we only let M arcs iterate once between each call to D rift , andwe limit the temperature correction to half of what the code sug-gests. This way we stop the overheating of the atmosphere andallow the dust formation to react to the temperature change be-fore it becomes too large. When the temperature correction is be- low T ≤
10 K we consider the cloud layer stable and let M arcs converge fully without calling D rift again.
3. Results
We have created a small grid of models for late type M-dwarfsand early L-type brown dwarfs with e ff ective temperatures of T e ff = − T =
100 K. They all havesolar initial abundances and a surface gravity of log( g ) = . We present the temperature-pressure profiles of our models inFigure 1. Convection sets in at around P g > dyn / cm − and isthe predominant mode of energy transport in the bottom layersof the atmosphere. In the upper layers the temperature gradientis very shallow.In Figure 2 we compare the temperature-pressure profiles ofour cloud forming models with models where the cloud form-ing has been switched o ff . For T e ff < ff ect is so small in the beginning,that it barely a ff ects the structure of the model. At T e ff = ff ect in the outer layers. This happens because thedepletion of the gas phase elements that are now bound in dustgrains leads to a depletion of the gas phase molecules that aremade up of those specific elements. Although these moleculesare a small fraction of the overall number of molecules, someof them are important absorbers and their depletion significantlyreduces the opacity of the atmosphere. As long as the clouds arenot substantial enough for their own opacity to compensate forthe decreased molecular opacity, the a ff ected layers will cool alittle. At T e ff = −
20 K, at T e ff = T e ff = P g = − dyn / cm coincide with the lower and densest part of the formingclouds (see Figure 3).There is a general tendency for the temperature irregularitiesto shift downwards for decreasing e ff ective temperatures. Thiscan be explained by a combination of the thermal stability tem-perature moving downward, plus the withdrawal of the convec-tion zone and a lower velocity of the convective cells, whichmakes the element replenishment less e ff ective and causes theclouds to sink down a little into the atmosphere. Figures 3 and 4 present a more detailed view of how the di ff erentprocesses involved in cloud formation depend on and react toeach other as we move down through the atmosphere, as well ashow the size and composition of the cloud particles changes inresponse.Starting at the top of the atmosphere and moving down, thenucleation rate J ∗ rises quickly due to increasing collisional ratesas the density increases. When a distinct local temperature T θ ≈ Article number, page 4 of 17iana Juncher, U ff e G. Jørgensen and Christiane Helling: Atmosphere modeling with cloud formation Fig. 1.
The T - P g profiles for our grid of models with varying e ff ectivetemperatures, log( g ) = . Fig. 2.
The T - P g profiles for our grid of models with clouds (full drawnlines) compared to cloud free models (dashed lines). All models havelog( g ) = . ff ective temperature of the modelis. The nucleation rate causes a rise in the number density of dustgrains n d , and the peak coincides with the first rapid increase inthe number of dust particles.After the nucleation rate peaks the number density flattensout until it sharply increases again at the bottom of the cloudlayers as the result of gravitational settling and cloud particleaccumulation before complete evaporation. In the middle of thecloud layer, the cloud particle mass density ρ d keeps increasingwhile the number density does not, showing that while the nu-cleation of new cloud particles have stopped, the already existingones fall into deeper layers and are still growing larger. This co-incides with the growing volume fraction of cloud particles otherthan TiO [s]. The silicates Mg SiO [s], MgSiO [s] and SiO [s]are the first to condense on the seed particles, quickly followedby MgO[s] and Fe[s] and then finally Al O [s].At the bottom of the cloud layers the cloud particles evapo-rate at the high temperatures, causing a rapid decrease in theirmass density and a drop in average particle size.The net growth rate is χ net > χ net < Fig. 3.
The nucleation rate J ∗ , net growth rate χ net , mass density ρ d and number density n d of the dust grains, the mean grain size < a > , theconvective velocity v conv , and the drift velocity v d as a function of gaspressure. Color coding is indicated in panel 4. for the solids to e ff ectively condense on the nucleation parti-cles. It peaks before the nucleation rate indicating that it dependsmore on the amount of available surface area than on the forma-tion of new small particles. The mean grain size < a > is deter-mined by the net growth rate, and the first and second increasein the mean grain size happens in sync with the first and secondperiod of growth. Near the bottom of the cloud layers the netgrowth rate and mean grain size rapidly drops as the cloud par-ticles completely evaporate. The fluctuations in the net growthrate is due to the di ff erent solid species evaporating at di ff erenttemperatures.The drift velocity is initially decreasing as the gas density -and therefore the friction - increases. The decreasing ends whenthe second period of growth sets in, as the larger cloud particlescan more easily overcome the friction with the surrounding gas,because the downward accelerating force is proportional to grainsize cubed (i.e. the grain mass), while the upward restoring force(the friction) is proportional to grain size squared. The cloud particles are formed from the elements Mg, Si, Ti,O, Fe and Al in the present model. Figure 5 shows how theirabundances in the gas phase change as a function of atmosphericdepth as they are bound in cloud particles. In general, the more
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Fig. 4.
The volume fraction of each of the seven solid species in a dustgrain as a function of gas pressure. All fractions sum up to unity. Thecolor coding is the same as in Figure 3. rare elements are stronger depleted. While the large abundanceof O is barely a ff ected by the cloud formation, the other elementsare clearly depleted in the cloud forming regions. Since we useTiO [s] as the seed particle, the relatively small abundance ofTi is strongly depleted in the upper layers. The element deple-tion of the remaining elements sets in a little later when there isavailable seed particles for them to condense on. The depletionis largest where the nucleation peaks (compare Figures 5 and 3)and then decreases as the cloud particles reach the lower warmerlayers and start to evaporate. Because the elements rain out withthe cloud particles, we see an overabundance of condensable el-ements right below the cloud base, which is then transported up-ward with the gas convection. This will result in an increase ofthe corresponding gas opacity species. Based on the considerations in the previous section we can iden-tify di ff erent regions within the clouds that each have their owncharacteristics and dominant processes. The formation, growth,and evaporation of the cloud particles in a gas of a given chemi-cal composition is a function of temperature as well as gas pres-sure, and therefore to be understood as ”a race" between thechanging values of these two variables. Decreasing the tempera-ture will enhance cloud particle formation, as will increasing gaspressure. The changing conditions for a cloud particles during Fig. 5.
Relative element depletion in the gas phase as a function ofdepth. Color coding is the same as in Figures 3 and 4 and is also in-dicated with the legend in panel 4. its movement down through the photosphere, with its increasingtemperature and increasing gas density, is therefore determinedby the ratio of these two variables.With respect to the grain size distribution, we can divide theclouds into five distinct zones based on how the mean size of acloud particles changes as we move from the top to the base ofthe clouds. The five regions are illustrated for a model with T e ff = g ) = . / H] = Nucleation
At the top of the cloud layers the nucleation of gas moleculesis the dominant process, and the gas phase is therefore highlydepleted in Ti relative to all other elements.2.
First growth
As the cloud particles fall down into the atmospheric layers,the increasing density and element replenishment allow fora growing number of possible surface reactions on the smallseed particles, and the cloud particles increase considerablyin size. As a result, the gas becomes more and more depletedin the elements that make up the cloud particles. The rateof newly forming seed particles still increases in this region,but it is the rapid growth that has the dominant e ff ect on theaverage cloud particle size < a > .3. Drift
The increasing density of the gas combined with the increas-ing < a > causes the decent of the cloud particles to slow, Article number, page 6 of 17iana Juncher, U ff e G. Jørgensen and Christiane Helling: Atmosphere modeling with cloud formation Fig. 6.
The five regions of the clouds: I nucleation, II first growth, IIIdrift, IV second growth, V evaporation. which reduces the collision rate between the cloud particlesand the gas molecules. This will decrease the growth rate.In the same region the nucleation rate peaks and the averagecloud particle size remains constant, as the impeded growthof the large cloud particles is compensated for by the rapidformation of new small grains.4.
Second growth
When the nucleation rate suddenly drops, the growth in grainsize is no longer balanced by the formation of small cloudparticles, and the mean grain size therefore increases rapidlyagain. This ends the decrease of the drift velocity which re-mains more or less constant until the grains start evaporating.This and the still increasing density allow for an increase inthe net growth rate.5.
Evaporation
In the lowest layers of the cloud particles start to evaporate asthe temperature reaches the monomerization energies of thedi ff erent solids, and the mean grain size decreases, droppingvery fast as the last cloud particles evaporate at the cloudbase.We note that this stratification prevails as long as the hydro-dynamic time scales are longer than any of the time scales rep-resenting cloud formation processes.
4. Synthetic spectra
Detailed studies of the many complex physical and chemicalprocesses that take place in our model atmospheres are an impor-tant part of understanding and developing our theories of stars,but at the end of the day it is only the light that leaves the atmo-sphere, the emitted spectrum, that provides us with a way to di-rectly compare our stellar models with observations of real stars.
Figure 7 illustrates the impact of atomic (blue) and molecular(red) line absorption on the spectrum of a dust free model with T e ff = g ) = . / H] =
0. At such low e ff ectivetemperatures the absorption of atoms does not really a ff ect thestructure of the model, but they do create a few strong absorptionlines in the ultraviolet and visible part of the spectrum. The mostprominent are the two CaII lines at 3968 / / / / / Fig. 7.
Spectral contributions of gas opacity sources for a cloud-freeM arcs -model atmosphere for T e ff = g ) = .
5, [M / H] = Fig. 8.
Spectral contribution of dust (normalized with respect to thecontinuum flux) for models of decreasing e ff ective temperature. Still, it is the molecules that dominate the spectrum, completelyobscuring most of the atomic lines except in the ultraviolet re-gion. A more detailed look at the individual absorption of themolecules is presented in Appendix B.Figure 8 shows how the increasing dust opacity a ff ects thespectrum for models of decreasing e ff ective temperatures. Thenormalized flux (the flux divided by the continuum flux) includesonly the e ff ect of dust on the spectrum. The dust opacity in-creases in a broad band that covers the optical and near-infraredwavelength regions, peaking at around 1-3 µ m. For our coolestmodel, about half of the light is being blocked by the cloud lay-ers at λ ≈ µ m. This is similar to the e ff ect of water vapour inour cloud-free T e ff = ff ect of dust opacity (green) on thespectrum as a function of wavelength in comparison to the vari-ous gas opacity contributions (atoms – blue, molecules –red) fora cloud-forming D rift -M arcs -model atmosphere ( T e ff = g ) = . / H] = . arcs is shown. The dust extinction is most prominent in abroad band that covers the optical and near-infrared regions andpeaks at around λ = − µ m, where it is comparable to oreven greater than the molecular absorption. For λ > µ m thedust extinction has a noticeable dampening e ff ect on the molec-ular absorption bands, which would otherwise have completelydominated the spectrum. The two sharp peaks at short wave-lengths are numerical artifacts pointing to challenges with theMie calculations. They, however, occur in the ultraviolet part of Article number, page 7 of 17 & A proofs: manuscript no. aa
Fig. 9.
The individual and combined e ff ects of atomic opacity, molecu-lar opacity and cloud opacity on the normalized flux (total flux dividedby the continuum flux) of a cloud-forming D rift -M arcs -model atmo-sphere with T e ff = g ) = . / H] = the spectrum where the opacity is heavily dominated by atomicabsorption and therefore does not have an e ff ect on the spectrum(nor on the structure of the model). The dust extinction increaseswith decreasing e ff ective temperatures. The opacity of the atmosphere changes with wavelength andtherefore so does the optical depth τ ( λ ). We can determine theoptical depth of the atmosphere from our synthetic spectrum andthereby estimate how deep into the atmosphere we can see at aspecific wavelength. In Figure 10 we plot the total normalizedflux (top) and the geometrical depth, z ( λ ), where τ ( λ ) = T e ff = g ) = . / H] = > ≈
10% translates into a ∆ z ( λ ) ≈
50 km which is15% of the total geometrical extension of the atmosphere of alog( g ) = . Ultra cool dwarfs emit the majority of their radiation flux at near-infrared (NIR) wavelengths and their discovery and classifica-tion is therefore mainly conducted by NIR spectroscopic instru-ments. One such instrument is the SpeX spectrograph mountedon the 3 m NASA Infrared Telescope Facility, which providesmoderate and low resolution broad-band NIR spectra (Rayneret al. 2003). SpeX spectra has proved ideal for NIR classifica-tion, characterization of atmospheric and physical properties aswell as testing atmosphere models (Burgasser 2014), and is madereadily available from the online SpeX Prism Spectral Libraries that we have compared with our synthetic D rfit -M arcs spectra.The SpeX spectra are all normalized, have a resolution of R = λ/ ∆ λ ≈ λ ≈ . − . µ m. For a single comparison, we re-sampled our http: // / spexprism Figure 6.6.1:
The synthetic spectrum and optical depth of a model with = 2000 K, log( ) = 4 5 and
Fig. 10.
The total nominal flux (top) and the atmospheric depth, z ( τ λ =
1) for cloud-forming D rift -M arcs -model atmosphere for T e ff = g ) = .
5, [M / H] = τ ROSS ) = τ ROSS ) = −
10 is 333 km. synthetic spectrum to match the resolution and range of the ob-served spectrum and then fitted the synthetic spectrum to the ob-served spectrum by simply scaling the synthetic spectrum. Weused the non-linear least squares curve fitting routine MPFIT(Markwardt 2009) which identifies the best fit as the one withthe lowest value of χ = N (cid:88) i = ( A · f synth,i − f obs,i ) σ obs,i / ( N − , (1)where A is the scaling factor, the only free parameter, and N is the number of data points. We repeated this process for ev-ery combination of synthetic and observed spectrum, in the endidentifying the best fitting synthetic spectrum for each observedspectrum as the one with the lowest value of χ .Most of the χ values were in the range of χ ≈ . −
15 witha few very large exceptions. Since our grid serves as an indica-tion of the direction we are going in with our models, we expectthat a good deal of the fits will be considerably improved oncewe have computed a more complete grid that includes variationsin surface gravity or metallicity. We are therefore wary of sys-tematic o ff sets in our fits and only consider the best fit of a fewselected spectral sub classes, where χ < . In Figure 11 we present the comparison between the observedspectra of six M-dwarfs and our best fit models. The earliest sub-type that can be fitted by our models is M4.5. With an e ff ectivetemperature of T e ff = . − . µ m, only disturbed slightly by Article number, page 8 of 17iana Juncher, U ff e G. Jørgensen and Christiane Helling: Atmosphere modeling with cloud formation Table 5.
The name, spectral type and data reference for the observed spectra together with the parameters of our best fit model. All models havelog( g ) = . M / H ] = Object Best fit model
Name SpT Reference T e ff χ + + + / L0.5 Burgasser et al. (2008) 2500 K 2.222MASS J15500845 + + + + + µ m and CrH at 0 . − . µ m. The absorption bandin the model at λ = . µ m is a mix of CrH, TiO and FeH inorder of influence, but it is not observed in the spectra of thisstar. For λ > . µ m the broad absorption bands of H O becomethe main absorption features and they stay almost undisturbedby other molecules and atoms except at λ = . − . µ m whereCO absorption causes the small fluctuations.The absorption issomewhat underestimated at λ = . − . µ m for most of themodels and slightly overestimated at λ = . − . µ m. We notehowever that the deviation is not correlated with T e ff , and thatthe magnitude of the two deviations are not correlated with oneanother. We therefore conclude that the mismatch most likelyis due to chemical abundance e ff ects beyond the range of ourpresent grid.None of the M-dwarfs reach e ff ective temperatures below T e ff = ff ective temperatures decrease, the peak of their spec-trum shifts towards longer wavelengths. The intensity of the TiObands grows larger and are blended with the increasing absorp-tion of VO and CrH. The absorption of CaH also increases at λ = . − . µ m but has a very small e ff ect on the spectrum.The increase in absorption of CrH, VO and FeH at λ = . µ mis well matched by the models. Finally, the absorption of H O inthe infrared increases significantly as well, each band growingdeeper with decreasing e ff ective temperature. With the massivesuppression of the continuum due to the absorption of molecules,the coolest M-dwarfs are clearly far away from being ideal blackbody radiators. As demonstrated in Figure 2, clouds barely a ff ectthe atmosphere of objects with T e ff > In Figure 12 we present the comparison between the observedspectra of six L-dwarfs and our best fit models. The latest subtype that can successfully be fitted by the model grid in ourpresent work is L6. For later sub types χ becomes too largeas the e ff ective temperatures go below our lowest grid tempera-ture of T e ff = ff ective temperatures, theabsorption from 0 . − . µ m gradually becomes characterizedby equally strong TiO and VO bands. The absorption of CaHand CrH also becomes stronger in that region, but since theirbands tend to coincide with the stronger TiO and VO bands, theydo not a ff ect the spectrum that much. The absorption feature at λ = . µ m is the result of a peak in CrH absorption as well Fig. 11.
Mid- and late M-dwarf SpeX observations fitted with D rift -M arcs . as absorption from TiO and FeH. The other noticeable absorp-tion features at λ = . µ m is caused by the superposition of theabsorption peaks of CrH, H O, VO and CaH and FeH.The infrared part of the SpecX spectral region is dominatedby three strong absorption features at 1.4 µ m, 1.9 µ m and 2.5 Article number, page 9 of 17 & A proofs: manuscript no. aa µ m, and corresponding opacity minima at 1.6 µ m and 2.2 µ m.The intensity of the absorption bands are determined by a tem-perature, pressure and elemental abundance dependent combi-nation of CO and H O and could also include contributions byother species with yet incomplete opacity data. The flux at theintensity minima are to a large extend determined by the morecontinuum-like dust absorption. The slight mismatch betweenour synthetic spectra and the SpecX infrared observations couldtherefore be due to incomplete inclusion of a combination ofany of these factors, but it is probably more likely (since thereis no clear T e ff dependence on the quality of the fits) to be dueto the smallness of the grid of models yet, which does not al-low us to vary the chemical abundances and gravity su ffi cientlyfor more accurate fits to the observed spectra. The present pa-per is, however, not aiming either at determining the parametersof the presented objects by detailed spectral fitting, but rather todevelop the basic principles of incorporating self-consistent dustformation into the gas phase atmospheric computation, and testwhether such models relate realistically to observations. As suchFigures 11 and 12 fully serve their purpose of demonstrating thatthis has been achieved, and we will leave the detailed matchingto a future paper with a more extended grid.In general we see that the optical region is dominated byTiO and VO absorption bands, the near-infrared region by strongmetal hydride bands (CrH, FeH, and CaH), and the infraredregion by the broad, cloud-opacity dampened H O absorptionbands.We note that our best fit model to 2MASS J1416 is severalhundred Kelvin warmer than a typical L6 type dwarf. This par-ticular L dwarf has been identified as an unusually blue objectfor its spectral type (Bowler et al. 2010), and it is therefore likelythat non-solar metallicities or other e ff ects makes it impossiblefor our small grid of models to fit it correctly.Furthermore, 2MASSW J0320 might be an unresolved lateM + T dwarf binary system (Burgasser et al. 2008) and can inthat case not be fitted well by a single model spectrum.
Hot Jupiters have deep hydrogen-helium atmospheres. Some ofthem orbit so close to their parent stars that they have surfacetemperatures larger than T = rift -M arcs tomodel such an atmosphere where we do not yet take into accountthe irradiation by the host star.The atmosphere of WASP-19b was modeled by Andersonet al. (2013) using the spectral retrieval methods developed inMadhusudhan & Seager (2009, 2010, 2011), which utilize para-metric ( T gas , P gas ) structures in combination with a cloud-freegas made of H , H O, CO, CH , CO , and NH . In the retrievalapproach, the ( T gas , P gas ) structure and the molecular abundancesare fitting parameters used to retrieve the observed spectrum. Inthe D rift -M arcs approach, on the other hand, the ( T gas , P gas )structure and the abundances of the individual gas- and dust-species are computed from first principles self-consistently withthe radiative transfer, energy balance, opacities, and dust forma-tion, as described above. There are therefore no free parametersin D rift -M arcs spectrum simulations (but, as mentioned above,irradiation is not yet included in the version presented here), andthe best fit model gives direct information about the tempera-ture profile and the chemical composition of the planetary atmo-sphere.WASP-19b is a transiting exoplanet with a mass of M p = . J and a radius of R p = . J in a close orbit aroundits parent star with a period of only P = .
789 days, as deter-
Fig. 12.
Early- and late L-dwarf SpeX observations fitted with D rift -M arcs . Table 6.
The relative flux of the exoplanet WASP-19b with respect toits star at di ff erent wavelengths (Table 4, Anderson et al. (2013)). Wavelength F p / F (cid:63) Reference1 . µ m 0 . ± . . µ m 0 . ± . . µ m 0 . ± . . µ m 0 . ± . . µ m 0 . ± . . µ m 0 . ± . T e ff = g ) = / H] = rift -M arcs to compute a stellar model atmo-sphere with these parameters and calculated its synthetic spec-trum, the flux f (cid:63) . The relative flux F p / F (cid:63) which we receive on Article number, page 10 of 17iana Juncher, U ff e G. Jørgensen and Christiane Helling: Atmosphere modeling with cloud formation the Earth from the two is F p F (cid:63) = f p f (cid:63) (cid:32) R p R (cid:63) (cid:33) , (2)where ( R p / R (cid:63) ) = . ± . F p / F (cid:63) for each of our cloud forming D rift -M arcs models, setting f p as their respective fluxes. Comparingthese synthetic planet-to-star fluxes with the observed planet-to-star flux, we found that it is best described by our cloud forming,non-irradiated model with T e ff = g ) = / H] = g ),which therefore cannot be determined from the spectrum (log( g ) = g ) = ff ective temperature of the planet with a Planck function of thestellar e ff ective temperature, and fitting this ratio to the Spitzermeasurements. In this way they found values between 2260 and2750 K (depending on the filter they fitted to). They also esti-mated the planetary equilibrium temperature based on the knowne ff ective stellar temperature and the planetary orbital size, and byassuming a planetary albedo of zero. In this way they reached ef-fective temperatures of WASP-19b between 2040 K and 2614 K(or actually, between 2433 K and 3109 K when correcting fora missing √ T e ff as given inthe caption to their Table 3). The range in temperature reflectsa range in assumed e ffi ciency in energy transport from day- tonight-side of the planet. They also analysed whether a tempera-ture inversion was visible in the measured flux distribution, andconcluded that their temperature inversion profile was inconsis-tent with the observed flux distribution (however, seemingly withlacking the spectral features of the inversion in their computedspectra, due to lacking chemical equilibrium and relevant opac-ities in the computations). Our estimate of T e ff = / O ratio, albedo zero and no temperature in-version. This is quite encouraging, because it qualitatively pointsat an atmosphere with low albedo, that is relatively clear, absorbmost of the incoming energy in the bottom of the atmosphere(with winds that will transport energy to the backside, but not soe ffi cient that the planet reach equal day and night temperature),and has no sign of a strong temperature inversion (with the cau-tion that neither we nor Anderson et al really have analysed thee ff ect of a temperature inversion, due to the two di ff erent compu-tational limitations mentioned above). Figure 13 shows the com-parison of our synthetic spectrum and the Spitzer observations.
5. Cloud particle porosity
Material properties are an essential input for every model. Thechallenge of obtaining such input has recently been outlined in
Fig. 13.
The best-fit synthetic transit spectrum for WASP-19b forlog( g ) = . g ) = . rift -M arcs model atmosphere simulations for the star ( T e ff = g ) = / H] = T e ff = Fortney et al. (2016). Here we shortly discuss the e ff ect of poros-ity on the cloud opacity.Figure 14 shows that the porosity of cloud particles can havea considerable e ff ect on the opacity. We chose to represent thee ff ect in terms of local Planck mean opacities as this allows usto plot a meaningful measure of the opacity as a function of thewhole atmospheric extension. We compare the integrated opac-ity for one example cloud-forming model atmosphere ( T e ff = g ) = / H] = ff ect. Interestingly, wesee that by increasing the porosity slightly the cloud grains be-come more opaque. If the porosity is too high the opacity dropsagain since the light can pass unhindered through a large part ofthe cloud grains.Porosity could arise if the cloud particles do not attain a com-pact shape during their formation or evolution, but rather developfractal shapes instead. We are familiar with this process fromEarth’s atmosphere as "snow". Comets are examples of a dif-ferent kind of porosity. It is not clear whether potential porositycan sustain as the cloud particles fall into deeper atmosphericlayers where their frictional interaction with the gas increases,which then would lead to a compactification or break-o ff of dan-gling structures. A more realistic scenario for relatively hot at-mosphere could be that di ff erent materials evaporate at di ff erenttemperatures, while others remain thermally stable throughoutthe entire atmosphere.
6. Conclusions
The coming years and decades will see a substantial technologi-cal development that will make it possible to obtain direct spec-tra of increasing quality of nearby exoplanets. Reliable interpre-tation of such high quality spectra will require detailed complex
Article number, page 11 of 17 & A proofs: manuscript no. aa
Fig. 14.
Testing the e ff ect of porosity on the cloud opacity ( T e ff = g ) = / H] = self-consistent model atmospheres, which at the same time willmake it possible to reliably quantify such exciting features as po-tential biomarkers in the atmosphere, and hence open a route forthe first scientific discussions of possible life forms on nearbyextrasolar planets. With these long-term goals in mind we havetaken the first steps to combine two well tested computer codesfrom stellar atmospheric theory, namely the M arcs radiative andconvective equilibrium code for gaseous atmospheres and theD rift dust and cloud formation code. In combination, the D rift -M arcs code that we here have presented for the first time, is ableto compute self-consistent model atmospheres that can includeboth radiative-convective energy transport, the chemical equilib-rium between both gas and dust species, as well as cloud for-mation and cloud destruction. These are necessary ingredientsto compute self-consistent models of exoplanetary atmospheres,and the exercise serves a double purpose, namely to pave theway for D rift -M arcs self-consistent general exoplanetary mod-els and to increase the accuracy of the stellar models of the typeof stars, M- L- and T-type stars (and brown dwarfs), whose orbit-ing exoplanets we already today are able to obtain crude spectraof. M-, L-, and T-dwarfs are very attractive targets when search-ing for new exoplanets by indirect methods. Their relativelysmall mass and size provide stronger signals for detection withthe radial velocity, astrometry, and transit methods. An inher-ent problem of these exoplanet search methods is that the uncer-tainty of the properties of the host star propagates to the prop-erties of its planet. It is therefore crucial that the stellar modelslinking the observations of a star to its properties are as precise aspossible, and the ultra-cool dwarf stars are much more complexto model than their larger and hotter cousins, mainly becausetheir temperatures are low enough for mineral clouds to formin their atmospheres. We have demonstrated when and how themineral cloud formation starts to play a role for the atmosphericstructure of our models, and we have shown that emergent spec-tra based on our D rift -M arcs model atmospheres are in goodagreement with observed spectra for the whole range of spec-tral types from mid-type M-dwarfs to late-type L-dwarfs ( T e ff = rift -M arcs code is therefore alreadyin its present form a reliable tool to accurately determine the stel-lar parameters and hence improve the parameters of exoplanetsorbiting cool dwarf stars. Hot Jupiter exoplanets orbiting solar-type and warmer starsare themselves of comparable ( T e ff , log( g )) values to the ultra-cool dwarf stars, and one would expect them to have slow ortidally locked rotations. They will therefore to a large extend re-semble the ultra-cool dwarf stars, and can therefore to a first ap-proximation be modeled in the same way as these. Crude spectraor photometric data points can already today be obtained for afew hot Jupiter exoplanets by subtracting the stellar spectrumduring occultation from the stellar spectrum with the exoplanetin di ff erent phases (i.e. positions of its orbit). We therefore testedour computed D rift -M arcs synthetic spectra against photomet-ric data of the hot Jupiter WASP-19b obtained from the Spitzersatellite. We found good agreement between the observed pho-tometry and a spectrum based on a D rift -M arcs model with T e ff = rift -M arcs code than presented here,as will the modeling of even more Earth-like exoplanets. Acknowledgements.
We are thankful to University of St Andrews for hospital-ity and financial support toward DJ during an extended stay in 2014 as partof her PhD thesis work, where part of this work was done. ChH highlight fi-nancial support of the European Community under the FP7 by the ERC start-ing grant 257431. We greatly appreciate discussions with, and valuable com-ments from, B. Gustafsson and K. Lodders, and are thankful for an inspir-ing and thorough referee report from D. Homeier. This research has benefit-ted from the SpeX Prism Spectral Libraries, maintained by Adam Burgasser athttp: // pono.ucsd.edu / ∼ adam / browndwarfs / spexprism. References
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Article number, page 13 of 17 & A proofs: manuscript no. aa
Appendix A: Included atoms and molecules
The following atoms and molecules were included in thechemical equilibrium calculations in M arcs . Atoms (38):
H, He, Li, Be, B, C, N, O, F, Na, Mg, Al, Si, P, S, Cl, K, Ca, Sc,Ti, V, Cr, Mn, Fe, Ni, Cu, Ge, Br, Rb, Sr, Y, Zr, Nb, I, Ba, La,Ce, Nd.
Molecules (210): H − , H , H O, OH, CH, CO, CN, C , N , O , NO, NH, C H ,HCN, C H, HS, SiH, C , CS, SiC, SiC , NS, SiN, SiO, SO, S ,SiS, TiO, VO, ZrO, MgH, HF, HCl, CH , CH , CH , NH , NH ,C N , C N, CO , F − , AlF, CaF, CaF , MgOH, Al O, AlOH,AlOF, AlOCl, NaOH, Si C, SiO , H S, CS , AlCl, NaCl, KCl,KOH, CaCl, CaCl , CaOH, TiO , VO , LiH, LiO, LiF, LiCl,BeH , BeO, BeF, BeCl, BeCl , BeOH, BH, BH , BO, B O, BS,BF, BCl, HBO, HBO , C − , C − , C H , NO − , N H , N H , CN ,C N , NO , NO , N O, N O , HNO, HNO , HNO , HCNO,O − , O − , OH − , CO − , C O, HCO, H CO, F , FO, NaH, NaO,NaF, MgO, MgS, MgF, MgF , MgCl, MgCl , AlH, AlO, AlO ,AlS, AlF , AlCl , SI − , SiH , SiF, SiF , SiCl, SiCl , PH, PH ,PH , CP, NP, PO, PO , PS, PF, PF , PCL, COS, SO , S O, SO ,Cl − , Cl , CCl, CCl , CCl , CCl , ClO, ClO , Cl O, SCl, SCl ,HClO, CClO, KH, KO, KF, CaO, CaS, TiF, TiF , TiCl, TiCl ,VN, CrN, CrO, CrO , FeO, FeS, FeF, FeF , FeCl, FeCl , NiCl,CuO, CuF, CuCl, SrO, SrS, SrF, SrF , SrCl, SrCl , SrOH, ZrH,ZrN, ZrO , ZrF, ZrF , ZrCl, ZrCl , HI, BaO, BaS, BaF, BaF ,BaCl, BaCl , BaOH, NBO, C , C , TiH, CaH, FeH, CrH. Appendix B: Synthetic spectra decomposition
We provide a detailed decomposition of the gas-contributions inFigs. B.1–B.3. At the shortest wavelengths, SiO, H , and COare all very strong absorbers, with SiO being the most influen-tial from 1 . − µ m. OH also has a fairly strong absorption from2 . − . µ m, but it is obscured by the SiO absorption. NH makesa short appearance around 3.4 µ m. TiO absorption starts to growfrom 4 µ m and completely dominates the spectrum from 4 . − µ m, with a few exceptions; at 7.5 µ m, and 8.8 µ m, the absorp-tion of TiO weakens but is compensated for by the absorption ofVO and CrH, respectively. In fact, if there had been no TiO inthe atmosphere, the metallic hydrides would have provided mostof the absorption from 0 . − . µ m with CaH peaking at 0.68 µ m, CrH at 0.88 µ m and 1 µ m, FeH at 1 µ m, MgH at 0.51 µ m,SiH at 0.42 µ m and TiH at 0.53 µ m. ZrO also shows its strongestabsorption in this region. Finally, H O absorption shows up at1.1 µ m and completely dominates the spectrum in the infraredand beyond. LiH and NO absorption both have a negligible ef-fect on the spectrum because of their very low partial pressures.Even though the absorption coe ffi cient of CO is larger than thatof CO in the optical, the partial pressure of CO at these hightemperatures is less than a thousandth of the partial pressure ofCO, and its spectroscopic features are therefore almost imper-ceptible. CH, C , CN and HCN are barely present in oxygen-rich atmospheres at these temperatures, and their contribution tothe absorption is consequently negligible. These molecules will,however, become of interest if carbon is enhanced compared tothe solar C / O ratio.
Article number, page 14 of 17iana Juncher, U ff e G. Jørgensen and Christiane Helling: Atmosphere modeling with cloud formation Fig. B.1.
Spectral contributions of atomic and molecular opacity sources for a cloud-free M arcs -model atmosphere (of T e ff = g ) = λ = . − . µ m (top to bottom panel). Article number, page 15 of 17 & A proofs: manuscript no. aa
Fig. B.2.
Same as Figure B.1 but for λ = . − . µ m (top to bottom panel).Article number, page 16 of 17iana Juncher, U ff e G. Jørgensen and Christiane Helling: Atmosphere modeling with cloud formation Fig. B.3.
Same as Figure B.1 but for λ = . − µµ